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Small Schemes and Varieties of Minimal Degree by D Small Schemes and Varieties of Minimal Degree by D. Eisenbud, M. Green, K. Hulek, and S. Popescu Abstract: We prove that if X ⊂ Pr is any 2-regular scheme (in the sense of Castelnuovo-Mumford) then X is small. This means that if L is a linear space and Y := L ∩ X is finite, then Y is linearly independent in the sense that the dimension of the linear span of Y is 1 + deg Y . The converse is true and well-known for finite schemes, but false in general. The main result of this paper is that the converse, “small implies 2-regular”, is also true for reduced schemes (algebraic sets). This is proven by means of a delicate geometric analysis, leading to a complete classification: we show that the components of a small algebraic set are varieties of minimal degree, meeting in a particularly simple way. From r the classification one can show that if X ⊂ P is 2-regular, then so is Xred, and so also is the projection of X from any point of X. Our results extend the Del Pezzo-Bertini classification of varieties of minimal degree, the characterization of these as the varieties of regularity 2 by Eisenbud- Goto, and the construction of 2-regular square-free monomial ideals by Fr¨oberg. Throughout this paper we will work with projective schemes X ⊂ Pr over an algebraically closed field k. The (Castelnuovo-Mumford) regularity of X ⊂ Pr is a basic homological measure of the complexity of X and its embedding in Pr that gives a bound for the degrees of the generators of the defining ideal IX of X and for many other invariants. The only schemes of regularity 1 are the linear spaces; but no classification is known for projective schemes of regularity 2. In this paper we prove a structure theorem for reduced 2-regular schemes, show- ing that their irreducible components are varieties of minimal degree and charac- terizing how these components can meet. We also show that the reduced structure on any 2-regular scheme is 2-regular, and thus we obtain a complete description of the reduced structures on 2-regular schemes. (Since a high Veronese re-embedding of any zero-dimensional scheme is 2-regular, one cannot hope to characterize the isomorphism types of all 2-regular non-reduced schemes.) Before stating our results we review some basic notions. For any subscheme X ⊂ Pr we write span(X) for the smallest linear subspace of Pr containing X. Recall that every variety (≡ reduced irreducible scheme) X ⊂ Pr satisfies the condition (∗) deg(X) ≥ 1 + codim(X, span(X)) The authors are grateful to BIRS, IPAM, MSRI, the DFG and the NSF for hospitality and support during the preparation of this work. 1 (see for instance Mumford [1976, Corollary 5.13]). We say that the variety X ⊂ Pr has minimal degree (more precisely, minimal degree in its span) if equality holds. Surfaces of minimal degree were classified by Del Pezzo [1886], and the classification was extended to all dimensions by Bertini [1907] (see Eisenbud-Harris [1987] for a modern account): Theorem 0.1 A projective variety of minimal degree in its span is either a linear space, a quadric hypersurface in a linear space, a rational normal scroll, or a cone over the Veronese surface in P5. This classification was extended to equidimensional algebraic sets that are con- nected in codimension 1 – the ones for which “minimal degree” is a good notion – by Xamb´o[1981]. For more general algebraic sets it is not clear that there should exist any interesting generalization of the equality in (∗) above. Nevertheless, the notion of smallness is just such a generalization. Varieties of minimal degree were characterized cohomologically by Eisenbud- Goto [1984], and their result offers a way to generalize the hypothesis of the Del Pezzo-Bertini Theorem to all projective schemes. To state their result recall that X ⊂ Pr is said to have regularity d, or to be d-regular, in the sense of Castelnuovo- i Mumford, if the ideal sheaf IX satisfies H (IX (d − i)) = 0 for all i > 0, or equiv- alently, if the j-th syzygies of the homogeneous ideal IX are generated in degrees ≤ d + j for all j ≥ 0 (see Eisenbud-Goto [1984], or Eisenbud [2004] for a proof of the equivalence). Theorem 0.2 A variety X ⊂ Pr has minimal degree in its linear span if and only if X is 2-regular. An old argument of Lazarsfeld (see for instance [2004]), recently refined by Sidman [2002], Caviglia [2003], Eisenbud-Green-Hulek-Popescu [2004] and others, shows that if X ⊂ Pr is d-regular and Λ is a linear subspace such that X ∩ Λ has dimension 0, then X ∩ Λ is also d-regular. When d = 2 we can rephrase this geometrically: We say that a finite scheme Y ⊂ Pr is linearly independent if the dimension of the linear span of Y is 1 + deg Y . We say that a scheme X ⊂ Pr is small if, for every linear subspace Λ ⊂ Pr such that Y = Λ ∩ X is finite, the scheme Y is linearly independent. (An alternative definition of smallness by a more general property of intersections is given in Theorem 2.2.) Lazarsfeld’s argument gives: Proposition 0.3 Any 2-regular scheme X ⊂ Pr is small. Our main results are that the converse holds in the reduced case, and that small reduced schemes have a simple inductive classification. To state the classification, r we say that a sequence of closed subschemes X1,...,Xn ⊂ P is linearly joined if, for all i = 1, . , n − 1, we have (X1 ∪ ... ∪ Xi) ∩ Xi+1 = span(X1 ∪ ... ∪ Xi) ∩ span(Xi+1). 2 Theorem 0.4 Let X ⊂ Pr be an algebraic set. The following conditions are equivalent: (a) X is small. (b) X is 2-regular. (c) X = X1 ∪ ... ∪ Xn, where X1,...,Xn is a linearly joined sequence of varieties of minimal degree. The implication (c) ⇒ (b) is easy (see Proposition 3.1) while (b) ⇒ (a) is a special case of Proposition 0.3 (see also Section 1 for details). Most of this paper is occupied with the proof of the implication (a) ⇒ (c), which requires a delicate geometric analysis of the notion of smallness in the style of classical projective geometry. One of the things that makes the argument subtle is the fact that the linearly joined property of a sequence of varieties is strongly dependent on the ordering, as the following example shows. 4 Example 0.5 Let L0 be a line in P , and let L1,L2,L3 be 3 general lines that S3 meet L0. The union X = i=0 Li is 2-regular; in fact it is connected in codimension 1, has minimal degree, and is a degeneration of a rational normal quartic curve. As required by Theorem 0.4, X can be written as the union of a linearly joined sequence of varieties L0,L1,L2,L3, which are trivially of minimal degree in their spans. On the other hand, the reverse sequence L3,L2,L1,L0 is not linearly joined. Indeed, the subset Y = L3 ∪ L2 ∪ L1 is not 2-regular: since Y meets the line L0 in three points, the ideal of Y requires a cubic generator. It is easy to check that there is no enumeration of the components of X as a linearly joined sequence for which the reverse sequence is linearly joined. In the special case where X is a union of coordinate subspaces, the equivalence of parts (b) and (c) of Theorem 0.4 had been proved by Fr¨oberg [1985, 1988] as an application of Stanley-Reisner theory. Unions of coordinate spaces correspond to simplicial complexes. Using earlier results of Dirac [1961] and Fulkerson-Gross [1965], Fr¨oberg showed that a simplicial complex corresponds to a 2-regular set if and only if it is the clique complex of a chordal graph. (We reprove this and give a generalization in Eisenbud-Green-Hulek-Popescu [2004]. See also Herzog, Hibi, and Zheng [2003] for a related path to Dirac’s theorem.) The orderings described in part (c) of Theorem 0.4 are called perfect elimination orderings in this context. See Blair-Peyton [1993] for a survey. Properties that are easy to check for algebraic sets satisfying one of the condi- tions of Theorem 0.4 may be quite obscure for sets satisfying another. Theorem 0.4 has a number of surprising algebraic and geometric consequences based on this ob- servation: Corollary 0.6 If X ⊂ Pr is a 2-regular algebraic set, then the union of any two irreducible components of X is again 2-regular. 3 By Example 0.5 the same cannot be said of a union of three components. Proof. By Theorem 0.4, the irreducible components of X are of minimal degree in their spans. and thus also 2-regular by Theorem 0.2. From Proposition 3.1 we see that any 2 components are again linearly joined. The result follows by applying Theorem 0.4 once again. Corollary 0.7 Let X ⊂ Pr be a 2-regular algebraic set. If p ∈ X is a point and r−1 πp denotes the linear projection from p, then πp(X) ⊂ P is 2-regular. By Theorem 0.4 a similar statement holds with “small” in place of “2-regular”. Proof. If X1,...,Xn is a linearly joined sequence of varieties of minimal degree, then πp(X1), . , πp(Xn) is a linearly joined sequence of varieties of minimal degree. Theorem 0.4 completes the proof. r r Corollary 0.8 If X ⊂ P is 2-regular, then Xred ⊂ P is also 2-regular.
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